Is there any common denominator among them (i.e. qualifications to be an Elder Evil), or is the category just a random grab-bag of Big Bad Boss monsters?
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It's sortof a grab bag. In fluff terms, elder evils are very ancient and powerful beings from bygone eras. In mechanics terms, they are a subtype of creature with specific world-changing powers (signs) among other traits, from the book "elder evils".
They include some types of Obryith lords, Baatorians, quasi-deitys from the Far-Realm, the Draeden, and probably the God-Beasts from Violet Dawn (which IIRC no longer exist)